non-MOS

I've been wondering about scales which are generated
by a rank 2 temperament -- whose tones are connected by
the generators of that temperament -- but which are not
MOS/DE. Is there any reason to exclude them? Not that
anyone is *excluding* them, but what justification can
we give for the horagrams in Paul's paper, or for doing
what Keenan and I both did -- look for good temperaments
with small MOSs without gaps?

Paul's "hypothesis" states that Fokker blocks that are
Constant Structures with all but one comma tempered out
are MOS/DE, and perhaps also the reverse. What are
these non-MOS/DE scales, then?

-Carl

On 6/8/06, Carl Lumma <ekin@lumma.org> wrote:
> I've been wondering about scales which are generated
> by a rank 2 temperament -- whose tones are connected by
> the generators of that temperament -- but which are not
> MOS/DE. Is there any reason to exclude them? Not that
> anyone is *excluding* them, but what justification can
> we give for the horagrams in Paul's paper, or for doing
> what Keenan and I both did -- look for good temperaments
> with small MOSs without gaps?
>
> Paul's "hypothesis" states that Fokker blocks that are
> Constant Structures with all but one comma tempered out
> are MOS/DE, and perhaps also the reverse. What are
> these non-MOS/DE scales, then?

Well, the piece I'm working on uses about 10 or 11 succesive pitches
from the Keemun temperament, so I'm not limiting myself to a MOS, but
I find myself thinking in terms of different 7-note MOSs I'm
modulating between. The 8-note scale doesn't really sound like a
coherent whole to me; it sounds like there's an extra note thrown in,
like a C major scale with an extra Bb. There's nothing wrong with it,
but I just can't conceive of it as a building block.

I guess that's not really much of a mathematical explanation... =P

Keenan

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Paul's "hypothesis" states that Fokker blocks that are
> Constant Structures with all but one comma tempered out
> are MOS/DE, and perhaps also the reverse. What are
> these non-MOS/DE scales, then?

Scales with the trivalent property might be a place to look. That's
what Scala calls it, and I think it means all the interval classes
have three "specific intervals". Also scales with two or three
elements per class. That sort of stuff is mostly what I was getting in
the proper scale search when I wasn't getting MOS.

Carl Lumma wrote:
> I've been wondering about scales which are generated
> by a rank 2 temperament -- whose tones are connected by
> the generators of that temperament -- but which are not
> MOS/DE. Is there any reason to exclude them? Not that
> anyone is *excluding* them, but what justification can
> we give for the horagrams in Paul's paper, or for doing
> what Keenan and I both did -- look for good temperaments
> with small MOSs without gaps?

The natural reason to exclude such scales is that they'll naturally be thought of as either a superset or subset of a nearby MOS. Why call something a "scale" after all? It suggests some kind of conceptual unity. Because an MOS is the most regular scale, it's going to have the most such unity. The caveat here is that hexachords were important historically. Perhaps that only worked becasue they weren't thought of as octave repeating. At least, looking at the historical significance of hexachords may be helpful in answsering you questions.

There are a lot of reasons for looking for an MOS/DE:

- You can transpose by the generator, and the number of notes that change is equal to the period. Actually, though I think this is true of all generated scales.

- There's always some tuning that's either proper or strictly proper. So if you're looking for proper scales the MOS are a good place to start.

- Other generated scales are likely to be improper in really ugly way, with ambiguous intervals all over the shop. They aren't easy to think about.

- An MOS/DE is good for notation. You can write any note in a rank 2 temperament in a unique way relative to the nominals as an MOS/DE. If you break uniqueness, that strikes me as ugly.

- With black/white keyboards, it's useful to think of them as having three different regular scales.

For generalized or split-key keyboards, there's nothing special about the actual number of keys being an MOS. I haven't noticed any tendency for keyboards from the split-key era to gravitate towards 19 keys to the octave.

> Paul's "hypothesis" states that Fokker blocks that are
> Constant Structures with all but one comma tempered out
> are MOS/DE, and perhaps also the reverse. What are
> these non-MOS/DE scales, then?

At least a periodicity block will always have the same number of notes as the relevant MOS. Some important non-MOS scales (quartertone rast and the pentachordal diatonics) have only two step sizes. Looking for an MOS/DE is a good way of finding such scales, because all you do is move a few notes around. The complexity is the same.

If you aren't planning to modulate, there's nothing special about generated scales. I'm guessing that most significant non-MOS/DE scales will prove not to be generated either. The only exception I can think of is the schismatic interpretation of the 22 srutis.

Graham

At 02:02 AM 6/8/2006, you wrote:
>Carl Lumma wrote:
>> I've been wondering about scales which are generated
>> by a rank 2 temperament -- whose tones are connected by
>> the generators of that temperament -- but which are not
>> MOS/DE. Is there any reason to exclude them? Not that
>> anyone is *excluding* them, but what justification can
>> we give for the horagrams in Paul's paper, or for doing
>> what Keenan and I both did -- look for good temperaments
>> with small MOSs without gaps?
>
>The natural reason to exclude such scales is that they'll naturally be
>thought of as either a superset or subset of a nearby MOS. Why call
>something a "scale" after all? It suggests some kind of conceptual
>unity. Because an MOS is the most regular scale, it's going to have the
>most such unity. The caveat here is that hexachords were important
>historically. Perhaps that only worked becasue they weren't thought of
>as octave repeating. At least, looking at the historical significance
>of hexachords may be helpful in answsering you questions.

Hey, that's true, I'd forgotten about hexachords. Too bad I don't
know anything about them.

>There are a lot of reasons for looking for an MOS/DE:
>
>- You can transpose by the generator, and the number of notes that
>change is equal to the period. Actually, though I think this is true of
>all generated scales.

I think so too.

>- There's always some tuning that's either proper or strictly proper.
>So if you're looking for proper scales the MOS are a good place to start.

I think this should be true of all generated scales also.

>- Other generated scales are likely to be improper in really ugly way,
>with ambiguous intervals all over the shop. They aren't easy to think
>about.

Sometimes, though, as with hanson, hanson[8] is proper and
the MOS hanson[7] is improper.

>- An MOS/DE is good for notation. You can write any note in a rank 2
>temperament in a unique way relative to the nominals as an MOS/DE. If
>you break uniqueness, that strikes me as ugly.

There may be something to this.

>> Paul's "hypothesis" states that Fokker blocks that are
>> Constant Structures with all but one comma tempered out
>> are MOS/DE, and perhaps also the reverse. What are
>> these non-MOS/DE scales, then?
>
>At least a periodicity block will always have the same number of notes
>as the relevant MOS. Some important non-MOS scales (quartertone rast
>and the pentachordal diatonics) have only two step sizes. Looking for
>an MOS/DE is a good way of finding such scales, because all you do is
>move a few notes around. The complexity is the same.

I guess my question was, in what sense do we consider a scale as
coming from a temperament. Does it matter if it's MOS? Does it
matter if the generators are continuous on a chain?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> I've been wondering about scales which are generated
> by a rank 2 temperament -- whose tones are connected by
> the generators of that temperament -- but which are not
> MOS/DE.

One way to study them might be in terms of what you might call the
transpositions graph. Take transpositions of the original scale, and
look at the set intersections. If for two transpositions the
intersection is nonempty, and there is no third transposition such
that the intersection is contained in it also, then draw a graph edge
between the two nodes representing the two transpositions.

If you do this with MOS, you get a chain graph. If you do it with a rank
3 temperament where one generator is an octave, you should get a
connected planar graph it seems to me.

>If you do it with a rank 3 temperament where one generator is
>an octave, you should get a connected planar graph it seems to me.

Are you saying non-MOS rank-2-based scales rank 3 temperaments?
I suppose that would make sense, since there aren't just 2 step
sizes anymore.

-Carl

Carl Lumma wrote:

>>- You can transpose by the generator, and the number of notes that >>change is equal to the period. Actually, though I think this is true of >>all generated scales.
> > I think so too.
> >>- There's always some tuning that's either proper or strictly proper. >>So if you're looking for proper scales the MOS are a good place to start.
> > I think this should be true of all generated scales also.

Not strictly proper.

>>- Other generated scales are likely to be improper in really ugly way, >>with ambiguous intervals all over the shop. They aren't easy to think >>about.
> > Sometimes, though, as with hanson, hanson[8] is proper and
> the MOS hanson[7] is improper.

That's interesting, and perhaps you could do something with it. But hanson[8] is, what, 36% stable? That's supposed to make it difficult to use as a proper scale.

>>- An MOS/DE is good for notation. You can write any note in a rank 2 >>temperament in a unique way relative to the nominals as an MOS/DE. If >>you break uniqueness, that strikes me as ugly.
> > There may be something to this.
> > > >>>Paul's "hypothesis" states that Fokker blocks that are
>>>Constant Structures with all but one comma tempered out
>>>are MOS/DE, and perhaps also the reverse. What are
>>>these non-MOS/DE scales, then?
>>
>>At least a periodicity block will always have the same number of notes >>as the relevant MOS. Some important non-MOS scales (quartertone rast >>and the pentachordal diatonics) have only two step sizes. Looking for >>an MOS/DE is a good way of finding such scales, because all you do is >>move a few notes around. The complexity is the same.
> > > I guess my question was, in what sense do we consider a scale as
> coming from a temperament. Does it matter if it's MOS? Does it
> matter if the generators are continuous on a chain?

It doesn't have to be a continuous chain, although it may matter in some cases. A scale belongs to a temperament if it can support a range of tunings within that temperament class.

Graham

GLOSSARY:

Proper - no intervals of n scale steps are bigger than intervals of n+1 scale steps

Strictly proper - all intervals of n scale steps are smaller than intervals of n+1 scale steps

Stability - the proportion of intervals of n scale steps that are a different size to all intervals spanning a different number of scale steps. Assumes propriety but not strict propriety.

MOS - a generated scale (abstraction of the spiral of fifths) with two step sizes